The concentration s(x, t) of pollutants, in a one-dimensional reservoir at position x and time t satisfies the diffusion equation
$${{\partial s(x,t)} \over {\partial t}} = D{{{\partial ^2}s(x,t)} \over {\partial {x^2}}}$$
on the domain 0 $$\le$$ x $$\le$$ L, where D is the diffusion coefficient of the pollutants. The initial condition s(x, 0) is defined by the step-function shown in the figure.
The boundary conditions of the problem are given by $${{\partial s(x,t)} \over {\partial x}}$$ = 0 at the boundary points x = 0 and x = L at all times. Consider D = 0.1 m2/s, s0 = 5 $$\mu$$mol/m and L = 10 m. The steady-state concentration $$\overline s \left( {{L \over 2}} \right) = s\left( {{L \over 2},\infty } \right)$$ at the center x = $${{L \over 2}}$$ of the reservoir (in $$\mu$$mol/m) is __________. (in integer)
Two discrete spherical particles (P and Q) of equal mass density are independently released in water. Particle P and particle Q have diameters of 0.5 mm and 1.0 mm, respectively. Assume Stoke's law is valid.
The drag force on particle Q will be ___________ times the drag force on particle P. (round off to the nearest integer)
A hydraulic jump takes place in a 6 in wide rectangular channel at a point where the upstream depth is 0.5 m (just before the jump). If the discharge in the channel is 30 m3/s and the energy loss in the jump is 1.6 m, then the Froude number computed at the end of the jump is _________. (round off to two decimal places)
(Consider the acceleration due to gravity as 10 m/s2.)
Water is flowing in a horizontal, frictionless, rectangular channel. A smooth hump is built on the channel floor at a section and its height is gradually increased to reach choked condition in the channel. The depth of water at this section is y2 and that at its upstream section is y1. The correct statements for the chocked and un-chocked conditions in the channel is/are